PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Statistical mechanics and thermodynamics → Statistical mechanics ↓


Fluctuations and correlations


In the field of statistical mechanics, the concepts of fluctuations and correlations play a vital role. The subtle dance between order and chaos is something that both these phenomena encapsulate beautifully, shedding light on the behaviour of systems at the microscopic level. This complex interplay is fundamental to our understanding of states of matter, phase transitions, and many other phenomena in physics.

Understanding the ups and downs

In statistical mechanics, fluctuation refers to a temporary change in a given property of a system. Since systems are composed of a large number of particles, properties such as energy, magnetism, density and others are not constant but rather oscillate around their average value.

Consider the energy of a system consisting of a large number of particles. Because of the continual exchange of energy between particles, the total energy is not constant, but rather fluctuates around an equilibrium value. These fluctuations become smaller relative to the average value as the number of particles increases, making them more prevalent in smaller systems.

Mathematical representation of fluctuations

Fluctuations can be described quantitatively using statistical measures. For a given observable property ( X ), the average or mean value ( langle X rangle ) is given by:

[ langle X rangle = frac{1}{N} sum_{i=1}^{N} X_i ]

where ( X_i ) denotes the value of the property for the ( i )-th particle, and ( N ) is the number of particles.

The fluctuation of (X) is characterized by its variance and is defined as:

[ sigma^2_X = langle (X - langle X rangle)^2 rangle ]

The standard deviation, ( sigma_X ), is the square root of the variance and provides a measure of the extent of fluctuations around the mean.

Exploring correlations

Correlations describe how much one part of a system affects or interacts with another part. This concept is important when considering interactions between particles, which can affect how observables such as magnetism or density fluctuate in a system.

Correlation in practice

Suppose we want to study the correlation between two observables, ( A ) and ( B ). The correlation function ( C(A, B) ) is defined as:

[ C(A, B) = langle (A - langle A rangle)(B - langle B rangle) rangle ]

This correlation function tells us how the deviations from their respective mean for ( A ) and ( B ) are related. A positive value indicates that an increase in ( A ) is associated with an increase in ( B ); a negative value suggests the opposite.

Looking at fluctuations and correlations

Example: Gas molecules in a container

Imagine a container of gas in which the particles are moving around randomly. The density of the gas particles in a small region of space may deviate from the average density due to the random motion. The density at one point in the container affects the density at nearby points because the molecules interact at short distances.

Correlation

The connected circles in our illustration represent gas molecules, and their combination shows correlations between position and motion from one place to another.

Example: Ising model

Consider the Ising model on a lattice, where each site can take a value of +1 or -1, representing magnetic spin directions. The correlation between the spins at different sites, (i) and (j), is an essential part of understanding the phase transition in this model. The correlation is captured by:

[ langle s_i s_j rangle - langle s_i rangle langle s_j rangle ]
Spin

Here, we denote lattice sites with spins indicated by arrows. The interactions between neighboring spins exhibit alignment and correlations that affect the system behavior.

Applications of fluctuations and correlations

Fluctuations and correlations have applications in various fields:

  • Phase transitions: Near the critical point of a phase transition, such as the change from a liquid to a gas, fluctuations increase, and correlations spread over longer distances.
  • Materials science: The mechanical properties of materials depend on atomic and molecular correlations that determine how structure and defects affect behavior.
  • Biophysics: In biological systems, understanding how fluctuations in molecular concentrations and interactions affect cellular behavior is fundamental to the modeling of cell processes.

Text example: Fluctuations in the financial markets

In financial markets, asset prices exhibit fluctuations similar to molecules in a gas. Here, fluctuations can indicate a change in market trends or volatility. Correlations between different stocks or bonds can affect investment strategies, just as spin interactions can affect magnetic properties in materials.

Conclusion

The concepts of fluctuations and correlation are cornerstone components in the study of statistical mechanics, which investigates the behavior of complex systems. From understanding the microscopic basis of thermodynamic phenomena to informing the design and discovery of new materials, these concepts extend the reach of theoretical physics into practical areas.

Through quantitative measurements such as variance, standard deviation, and correlation functions, and insightful visual and abstract models such as the Ising model, these phenomenologies capture the rich fabric of interactions and variations that define the physical world.


PHD → 4.3.4


U
username
0%
completed in PHD

← Prev (4.3.3)
Bose–Einstein and Fermi–Dirac statistics
Next (4.4) →
Phase transition

Comments 



Fluctuations and correlations

https://www.physicsflow.com/phd/4.3.4?pfal=en